The distance of the distribution of \(T_n\)  to a normal distribution has been investigated in various forms such as Berry–Esseen bounds [1]}, [2]}, [3]}, [4]} and non-uniform Berry–Esseen bounds [5]}, [6]}, [7]}, [8]}, to mention just a few. For an extensive survey on further self-normalized limit theorems we refer to [9]}.
